To determine the minimum number of shirts the retailer needs to sell in order to cover all its monthly costs, we need to find the break-even point where total costs equal total revenue.
The retailer has a fixed monthly cost of \[tex]$500 for keeping the online shop active and updated, a marginal cost of \$[/tex]5 per shirt, and a marginal benefit (selling price) of \$10 per shirt.
To find out the break-even point, we must set the total costs equal to the total revenue.
Step-by-Step Solution:
1. Total Cost (C): This consists of the fixed monthly cost plus the variable cost (marginal cost per shirt times the number of shirts sold).
So, total cost \( C \) can be expressed as:
[tex]\[
C = 500 + 5 \times (\text{number of shirts})
\][/tex]
2. Total Revenue (R): This is the marginal benefit per shirt times the number of shirts sold.
Therefore, total revenue \( R \) can be expressed as:
[tex]\[
R = 10 \times (\text{number of shirts})
\][/tex]
3. Break-even Point: At the break-even point, total cost equals total revenue.
[tex]\[
500 + 5 \times (\text{number of shirts}) = 10 \times (\text{number of shirts})
\][/tex]
4. Solve for the Number of Shirts:
[tex]\[
500 + 5 \times (\text{number of shirts}) = 10 \times (\text{number of shirts})
\][/tex]
Move the term involving the number of shirts on the left-hand side to the right-hand side:
[tex]\[
500 = 10 \times (\text{number of shirts}) - 5 \times (\text{number of shirts})
\][/tex]
Simplify:
[tex]\[
500 = 5 \times (\text{number of shirts})
\][/tex]
Divide both sides of the equation by 5:
[tex]\[
\text{number of shirts} = \frac{500}{5}
\][/tex]
[tex]\[
\text{number of shirts} = 100
\][/tex]
So, the retailer needs to sell 100 shirts in a month to break even and cover all its costs.
Answer: B. 100
